Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₃λ₁₂-σ₂₃,
σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}
{1} ⊥ {2} | {3}

Instrumental Variable:

3 | {}

Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₂λ₁₃-σ₂₃,
σ₁₁λ₁₃-σ₁₃

Single Door Criterion:

{1} ⊥ {3} | {}
{1} ⊥ {3} | {2}

Instrumental Variable:

2 | {}

Parameter: λ₁₄ Identified

Algebraic Equations:

σ₁₃²λ₁₄-σ₁₁σ₃₃λ₁₄+σ₁₄σ₃₃-σ₁₃σ₃₄,
σ₁₁σ₁₃σ₂₃λ₁₄-σ₁₁σ₁₂σ₃₃λ₁₄+σ₁₂σ₁₄σ₃₃-σ₁₁σ₂₃σ₃₄,
σ₁₁²σ₂₃²λ₁₄-σ₁₁σ₁₂²σ₃₃λ₁₄+σ₁₂²σ₁₄σ₃₃-σ₁₁σ₁₂σ₂₃σ₃₄

Single Door Criterion:

{1} ⊥ {4} | {3}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₃σ₂₃λ₃₄-σ₁₄σ₂₃-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄,
σ₁₃²λ₃₄-σ₁₃σ₁₄-σ₁₁σ₃₃λ₃₄+σ₁₁σ₃₄,
-σ₁₂σ₁₄σ₂₃+σ₁₁σ₂₃²λ₃₄-σ₁₂²σ₃₃λ₃₄+σ₁₂²σ₃₄

Single Door Criterion:

{3} ⊥ {2,4} | {1}
{3} ⊥ {4} | {1,2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₁₃σ₂₂+σ₁₂σ₂₃+σ₁₃ω₂₂,
σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₂₄ Identified

Algebraic Equations:

σ₁₄σ₂₃-σ₁₃σ₂₄+σ₁₃ω₂₄,
σ₁₂σ₁₄-σ₁₁σ₂₄+σ₁₁ω₂₄

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃,
σ₁₃²-σ₁₁σ₃₃+σ₁₁ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

-σ₁₄²σ₃₃+2σ₁₃σ₁₄σ₃₄-σ₁₁σ₃₄²-σ₁₃²σ₄₄+σ₁₁σ₃₃σ₄₄+σ₁₃²ω₄₄-σ₁₁σ₃₃ω₄₄,
-σ₁₂σ₁₄²σ₃₃+2σ₁₁σ₁₄σ₂₃σ₃₄-σ₁₁σ₁₂σ₃₄²-σ₁₁σ₁₃σ₂₃σ₄₄+σ₁₁σ₁₂σ₃₃σ₄₄+σ₁₁σ₁₃σ₂₃ω₄₄-σ₁₁σ₁₂σ₃₃ω₄₄,
-σ₁₂²σ₁₄²σ₃₃+2σ₁₁σ₁₂σ₁₄σ₂₃σ₃₄-σ₁₁σ₁₂²σ₃₄²-σ₁₁²σ₂₃²σ₄₄+σ₁₁σ₁₂²σ₃₃σ₄₄+σ₁₁²σ₂₃²ω₄₄-σ₁₁σ₁₂²σ₃₃ω₄₄

Total Effect of X₁ on X₄: λ₁₃λ₃₄+λ₁₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}

Vanishing Ideal

Constraint Set:

σ₁₂σ₁₃-σ₁₁σ₂₃

All constraints are determinantal

Trek-separation:

{1,2} ⊥ {1,3} | ({1},{})
{1,2} ⊥ {1,3} | ({},{1})